1. Sec 6.1 Median

2. Objective
Identify and construct medians in triangles

3. Medians
Picture:
Both sides are congruent
Median
vertex to midpoint

4. How many medians can a triangle have?
vertex to midpoint

5. 4/28/2017
Midpoint-
If you have a midpoint- then the segments on both sides are CONGRUENT!
That is why you will see the “tick marks”
For the measure of the entire line- add both sides!
EQ: What are the differences between medians, altitudes, and perpendicular bisectors?

6. M D P C 9 N
1. What is NC if NP = 18?
2. If DP = 7.5, find MP. 15

7. 14 18 6 A B C D E 1.What is ED if DC = 14? 2.What Is AC if BC is 9?
You Try the Following: A B C D 14 E
1.What is ED if DC = 14?
2.What Is AC if BC is 9? 18
3.If BC = 3, find AC. 6

8. So if you are given the length of the entire side, how do you find a missing segment?
If you are given the length of a segment, how to you find the entire side?
If you have a median- what do you know about each side?

9. A E B D C
If CD = 2x + 5, BD = 4x – 1, and AE = 5x –2, find BE.
BD = CD
AE = BE
BE = 13
4x – 1= 2x + 5
BE = 5x – 2
BE = 5(3) – 2
2x = 6
x = 3

10. The intersection of the medians is called the CENTROID.
Draw the Picture:
How many medians does a triangle have?

11. Also called, 'center of gravity’ or 'center of mass'
4/28/2017
Refer to the figure on the right. Imagine you have a triangular metal plate, and try and balance it on a point - say a pencil tip.
-Once you have found the point at which it will balance, that is the centroid.
Also called, 'center of gravity’ or 'center of mass'
EQ: What are the differences between medians, altitudes, and perpendicular bisectors?

12. Concurrent:
When three or more lines or segments meet at the same point.

13. Draw Picture
Theorem 6-1
Distance from vertex to centroid is twice the distance from centroid to midpoint. 2x x

14. Vertex to Centroid  LONGER (2x)
4/28/2017
Vertex to Centroid  LONGER (2x)
Centroid to Midpoint  shorter (x)
x + 2x = 3x = whole median
EQ: What are the differences between medians, altitudes, and perpendicular bisectors?

15. C
How much is CX? D
CX = 2(XF) E X
CX = 2(13) 13 B A F
CX = 26

16. Quick Assessment What is a median? What is a centroid?
4/28/2017
Quick Assessment
What is a median?
What is a centroid?
What does concurrent mean?
What is the vertex?
What is a midpoint?
EQ: What are the differences between medians, altitudes, and perpendicular bisectors?

17. Centroid Lab
Lab- With partners
Worksheet

18. C
How much is XD? D
AX = 2(XD) E X 18
18 = 2(XD) B A F
9 = XD

19. In ABC, AN, BP, and CM are medians.
4/28/2017
Ex: 1
In ABC, AN, BP, and CM are medians.
If EM = 3, find EC. C N
EC = 2(3) P E
EC = 6 B M A
EQ: What are the differences between medians, altitudes, and perpendicular bisectors?

20. In ABC, AN, BP, and CM are medians.
4/28/2017
Ex: 2
In ABC, AN, BP, and CM are medians. C
If EN = 12, find AN.
AE = 2(12)=24 N P E
AN = AE + EN B
AN = M A
AN = 36
EQ: What are the differences between medians, altitudes, and perpendicular bisectors?

21. Warm-UP
With an index card write one positive thing about me/my teaching/ etc.

22. Sec 6.2 Altitudes and Perpendicular Bisectors

23. Objectives
Identify and construct altitudes and perpendicular bisectors in triangles

24. vertex to opposite side and perpendicular
Altitude
Picture:
Altitude
vertex to opposite side and perpendicular

25. The altitude is the “true height” of the triangle.
4/28/2017
Altitude
The altitude is the “true height” of the triangle.
EQ: What are the differences between medians, altitudes, and perpendicular bisectors?

26. The altitude is the “true height” of the triangle.
Tell whether each red segment is an altitude of the triangle.
The altitude is the “true height” of the triangle.
YES NO
YES

27. The altitude is the “true height” of the triangle.
Tell whether each red segment is an altitude of the triangle.
The altitude is the “true height” of the triangle.
Draw examples page 235

28. Perpendicular Bisector
Both sides are congruent- make sure you see this or it is NOT a perpendicular bisector
Picture:
Perpendicular Bisector
midpoint and perpendicular

29. Tell whether each red segment is an perpendicular bisector of the triangle.NO NO
YES

30. Can you have both?
Can it be both an altitude and perpendicular bisector?
Help Me Draw Examples:

31. 4/28/2017
Quick Assessment
What is the difference between altitude and perpendicular bisector?
What is an altitude?
What is a perpendicular bisector?
How can the segment be both- altitude and perpendicular bisector?
EQ: What are the differences between medians, altitudes, and perpendicular bisectors?

32. Graphic Organizer Compare and Contrast: Perpendicular Bisector
Altitude
Median
Angle Bisector

33. Medians
Picture:
Both sides are congruent
Median
vertex to midpoint

34. Draw Picture
Theorem 6-1
Distance from vertex to centroid is twice the distance from centroid to midpoint. 2x x

35. Vertex to Centroid  LONGER (2x)
4/28/2017
Vertex to Centroid  LONGER (2x)
Centroid to Midpoint  shorter (x)
x + 2x = 3x = whole median
EQ: What are the differences between medians, altitudes, and perpendicular bisectors?